
where \(\alpha _{eff}=\alpha +\xi \,v^2\)  since \(v_{eff}=v\) . Note that \(A_0(r)=B_0(r)\)  stay constant at unity for all \(r\)  (this represents the vacuum Minkowski spacetime). In contrast, \(A(r)\)  is unity at the origin \(r=0\)  but dips below unity as \(r\)  increases until it plateaus to a positive value \(D\)  at large radius \(R\) . The value of \(D\)  is obtained numerically. Recall that localized matter in \(2+1\)  dimensions yields an asymptotically Minkowski spacetime with an angular deficit [1]}. Asymptotically, \(A(r)=D\)  and the spatial part of the metric (REF ) becomes \(\frac{dr^2}{D} + r^2 d\theta ^2\) . If we define \(r_0=r/D^{1/2}\)  and \(\theta _0= D^{1/2}\, \theta \)  we obtain a manifestly flat metric \(dr_0^2 + r_0^2 \,d\theta _0^2\)  but with \(\theta _0\)  ranging now from 0 to \(2 \pi D^{1/2}\)  instead of \(2\,\pi \) . Since \(0<D<1\)  there is an angular deficit of
\(\delta = 2 \pi \, (1-D^{1/2})\,.\) 
